Isochron foliations and global bifurcations: A case study
James Hannam, Bernd Krauskopf, and Hinke M. Osinga
The basins of attraction of periodic orbits or focus equilibria of a given vector field are foliated by forward-time isochrons, defined as all initial conditions that synchronize under the flow with a point of a given phase on the periodic orbit. Similarly, backward-time isochrons of repelling periodic orbits or focus equilibria foliate their respective basins of repulsion. We present here a case study of a planar system that features a sequence of bifurcations, including a saddle-node bifurcation of periodic orbits, a homoclinic bifurcation and Hopf bifurcations, that change the nature and existence of periodic orbits. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are curves in the plane, in regions of phase space where they exist simultaneously. Such tangencies are generically quadratic and associated with sharp turns of isochrons and phase sensitivity of the system. In contrast to the earlier reported cubic isochron foliation tangency (CIFT) mechanism, which generates a pair of tangency orbits, we find isochron foliation tangencies that occur along single specific orbits in the respective basin of attraction or repulsion. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, while a CIFT is not a associated with a topological change of the underlying phase portrait. The properties and interactions of isochron foliations are determined and illustrated by computing a representative number of forward- time and backward-time isochrons as arclength-parameterized curves with a boundary value problem setup; this approach to computing isochrons has been implemented in the MATLAB package CoCo.
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